Let $A$ be a $n\times n$-matrix with integers in the range $u..v$ , where $u<v$ are arbitary integers. Is there a formula, or at least, a good estimate, for the probability that the matrix is singular? In the case $u=v$, the answer is obviously $1$.
It is intuitively clear, that the probability very quickly tends to $0$, if the range increases. As I tried, the probability that a $4\times4$-matrix with integers from $-100..100$ is singular is very small indeed. Of course, brute force is hopeless, a simulation would give crude approximations. I prefer suitable bounds.